In Fig. 12-76, a uniform rod of mass m is hinged to a...

In Fig. $12-76,$ a uniform rod of mass $m$ is hinged to a building at its lower end, while its upper end is held in place by a rope attached to the wall. If
angle $\theta_{1}=60^{\circ},$ what value must angle $\theta_{2}$ have so
that the tension in the rope is equal to $m g / 2 ?$

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Key Concepts

Force Components

Forces acting at angles can be decomposed into perpendicular components. Resolving a force into its horizontal and vertical elements simplifies the analysis of static systems, such as determining how the tension in a rope contributes to the overall equilibrium of a hinged rod.

Geometric Relationships in Statics

Geometrical relationships, including the angles at which forces act and the dimensions of the components, are crucial in statics. These relationships allow for the correct identification of moment arms and the proper decomposition of forces, which are necessary for accurately applying the conditions of equilibrium.

Torque and Moment Arms

Torque is the rotational equivalent of force and is calculated as the product of a force and the distance from the pivot point at which the force is applied—the moment arm. Understanding how to compute torques is essential when balancing forces in systems, as seen when setting the torque due to gravity against the torque provided by the rope.

Rotational Equilibrium

In statics, rotational equilibrium means that the sum of the torques acting on a body about any pivot point must be zero. This ensures that the body does not start rotating, which is a necessary condition for static equilibrium in systems such as a rod held by a hinge and a rope.

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